In mathematics, the Binomial Inverse Theorem is useful for expressing matrix inverses in different ways.
If A, U, B, V are matrices of sizes p×p, p×q, q×q, q×p, respectively, then
provided A and B + BVA−1UB are nonsingular. Note that if B is invertible, the two B terms flanking the quantity inverse in the right-hand side can be replaced with (B−1)−1, which results in
This is the matrix inversion lemma, which can also be derived using matrix blockwise inversion.
Verification
First notice that
Now multiply the matrix we wish to invert by its alleged inverse
which verifies that it is the inverse.
So we get that—if A−1 and exist, then exists and is given by the theorem above.[1]
Special cases
If p = q and U = V = Ip is the identity matrix, then
Remembering the identity
we can also express the previous equation in the simpler form as
If B = Iq is the identity matrix and q = 1, then U is a column vector, written u, and V is a row vector, written vT. Then the theorem implies
This is useful if one has a matrix with a known inverse A−1 and one needs to invert matrices of the form A+uvT quickly.
If we set A = Ip and B = Iq, we get
In particular, if q = 1, then
See also
References
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